1 / 1

AC 0  ACC 0  PP, also TC 0  PP NC 1  PSPACE, … , NL  PSPACE

LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M. Faramawi, MBA Dr. Kenneth W. Regan.

agalia
Download Presentation

AC 0  ACC 0  PP, also TC 0  PP NC 1  PSPACE, … , NL  PSPACE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. LANDSCAPE OF COMPUTATIONAL COMPLEXITY Spring 2008 State University of New York at Buffalo Department of Computer Science & Engineering Mustafa M. Faramawi, MBA Dr. Kenneth W. Regan A complete language for EXPSPACE: PIM, “Polynomial Ideal Membership”—the simplest natural completeness level that is known not to have polynomial-size circuits. Arithmetical Hierarchy (AH) PIM n n TOT = {Me : Me is total, i.e. halts for all inputs} K(2) TOT EXPSPACE Succinct 3SAT Unknown but Commonly Believed: 2 2 RECRE • AC0  ACC0 PP, also TC0 PP • NC1 PSPACE, … , NL  PSPACE • P  EXP, NP  NEXP • PSPACE  EXPSPACE • L ≠ NL …………………….. L ≠ PH • P ≠ NP  co-NP ……… P ≠ PSPACE • NP ≠ p2  p2 ……… NP ≠ EXP NEXP co-NEXP = .REC = .REC K D nxn Chess D = {Turing machines Me: Me does not accept e} = the complement of K. (“Diagonal Language”) EXP RE co-RE Best Known Separations: REC For any fixed k, there is a problem in this intersection that can NOT be solved by circuits of size O(nk) QBF = .REC = .REC PSPACE BQP: Bounded-Error Quantum Polynomial Time. Believed larger than P since it has FACTORING, but not believed to contain NP. Polynomial Hierarchy (PH) p2 p2 WS5 p2 p2 L The levels of AH and PH are analogous, except that we believe NP  co-NP ≠ P and p2 p2 ≠ PNP, which stand in contrast to RE  co-RE = REC and 2 2 = RECRE BPP: Bounded-Error Probabilistic Polynomial Time. Many believe BPP = P. = poly.P = poly.P TAUT SAT NC1 NLIN NP co-NP TC0 QBF NTIME [n2] PARITY PP Probabilistic Polynomial Time PNP FACT NP PSPACE ACC0 MAJ-SAT CVP P AC0 P Analogy between Arithmetical and Polynomial Hierarchies C Low-Level Classes NP co-NP GAP D REG NL = poly.P AC0 ≠ ACC0 = poly.P UGAP p2 p2 L etc P WS5 DTIME [n3] co-NP NP DLIN REG Deterministic and Nondeterministic Time Hierarchies Within NP BQP FACTORING is not believed to be complete for BQP or for NP  co-NP. A DTIME [n2] REG ≠ L B Complexity “Main Sequence” FACT WS5, the word problem for the symmetric group S5, is a regular language that is complete for NC1 under AC0 many-one reductions. P BPP E Realm of Feasibility?

More Related